Motions of asteroids at the Kirkwood gaps: I. On the 3:1 resonance with Jupiter

Abstract

The behavior of asteroids at the 3:1 resonance is studied in a large parameter region, and the various motions that these asteroids show are summarized. First, the motions of asteroids are investigated qualitatively using a semianalytical method, and the variations of ( a, σ) and (e, ω) are obtained graphically for various values of the parameters (a = semimajor axis, σ = critical argument, e = eccentricity, ω = longitude of perihelion) . Next, to know the motions of asteroids more quantitatively, the orbital changes in asteroids are calculated by numerical integrations in the scheme of a planar elliptical restricted three-body problem. The main results are as follows: (1) The variations in a and σ are rather complicated, and the variations in e and ω depend strictly on the behavior of these elements. (2) The region where the resonance occurs is obtained on the a- e plane, and almost all numbered asteroids avoid this region. (3) The eccentricities of most asteroids in the resonance region change markedly ( Δe ⪆ 0.2 or much larger), and these changes take place in a period of 10 4–10 5 years. (4) In terms of the orbital changes of asteroids, the resonance region shows a complicated structure.